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Surfaces with filamentous structures are ubiquitous in nature on many different scales, ranging from forests to micrometre-sized cilia in organs. Hairy surfaces are elastic and porous, and it is not fully understood how they modify turbulence near a wall. The interaction between hairy surfaces and turbulent flows is here investigated numerically in a turbulent channel flow configuration at friction Reynolds number
$Re_{\unicode[STIX]{x1D70F}}\approx 180$
. We show that a filamentous bed of a given geometry can modify a turbulent flow very differently depending on the resonance frequency of the surface, which is determined by the elasticity and mass of the filaments. Filaments having resonance frequencies lower than the main frequency content of the turbulent wall-shear stress conform to slowly travelling elongated streaky structures, since they are too slow to adapt to fluid forces of higher frequencies. On the other hand, a bed consisting of stiff and low-mass filaments has a high resonance frequency and shows local regions of increased permeability, which results in large entrainment and a vast increase in drag.

Superhydrophobic surfaces are able to entrap gas pockets in between surface roughness elements when submerged in water. These entrapped gas pockets give these surfaces the potential to reduce drag due to the overlying flow being able to locally slip over the gas pockets, resulting in a mean slip at the surface. In this work we assess the separate effects that surface slip and surface texture have on turbulence over superhydrophobic surfaces. We show that the direct effect of surface slip does not modify the dynamics of the overlying turbulence, which remains canonical or smooth-wall like. The surface drag is governed by the difference between two virtual origins, the virtual origin of the mean flow and the virtual origin experienced by the overlying turbulence, in an extension of the theory from Luchini, Manzo & Pozzi (J. Fluid Mech., vol. 228, 1991, pp. 87–109) for riblets. Streamwise slip deepens the virtual origin of the mean flow, while spanwise slip deepens the virtual origin perceived by the overlying turbulence. Drag reduction is then proportional to the difference between the two virtual origins. We decompose the near-wall flow into background-turbulence and texture-coherent components, and show that the background-turbulence component experiences the surface as homogeneous slip lengths. The validity of the slip-length model can then be extended to larger texture size
$L^{+}$
than thought in previous studies. For
$L^{+}\gtrsim 25$
, however, we observe that a nonlinear interaction with the texture-coherent flow develops that alters the dynamics of the background turbulence, exhibiting a modified distribution of turbulent energy across length scales. This has the effect of reducing the velocity increment
$\unicode[STIX]{x0394}U^{+}$
compared to that predicted using homogeneous slip lengths and sets the upper limit of applicability of slip-length models.

The interaction of stationary streaks undergoing non-modal growth with modally unstable instability waves in a high Mach number boundary-layer flow is studied using numerical computations. The geometry and flow conditions are selected to match a relevant trajectory location from the ascent phase of the HIFiRE-1 flight experiment; namely, a
$7^{\circ }$
half-angle, circular cone with
$2.5$
mm nose radius, free-stream Mach number equal to
$5.30$
, unit Reynolds number equal to
$13.42~\text{m}^{-1}$
and wall-to-adiabatic temperature ratio of approximately
$0.35$
over most of the vehicle. This paper investigates the nonlinear evolution of initially linear optimal disturbances that evolve into finite-amplitude streaks, followed by an analysis of the modal instability characteristics of the perturbed, streaky boundary-layer flow. The investigation is performed with a stationary, full Navier–Stokes equations solver and the plane-marching parabolized stability equations (PSE), in conjunction with partial-differential-equation-based planar eigenvalue analysis. The overall effect of streaks is to reduce the peak amplification factors of instability waves, indicating a possible downstream shift in the onset of laminar–turbulent transition. The present study confirms previous findings that the mean-flow distortion of the nonlinear streak perturbation reduces the amplification rates of the Mack-mode instability. More importantly, however, the present results demonstrate that the spanwise varying component of the streak can produce a larger effect on the Mack-mode amplification. The analysis of planar and oblique Mack-mode waves modulated by the presence of the streaks shows that the planar Mack mode still dominates the instability characteristics of the flow. The study with selected azimuthal wavenumbers for the stationary streaks reveals that a wavenumber of approximately
$1.4$
times larger than the optimal wavenumber is more effective in stabilizing the planar Mack-mode instabilities. In the absence of unstable first-mode waves for the present cold-wall condition, transition onset is expected to be delayed until the peak streak amplitude increases to nearly 35 % of the free-stream velocity, when intrinsic instabilities of the boundary-layer streaks begin to dominate the transition process. For streak amplitudes below that limit a significant net stabilization is achieved, yielding a potential transition delay that can exceed 100 % of the length of the laminar region in the uncontrolled case.

A macroscopic boundary condition to be used when a fluid flows over a rough surface is derived. It provides the slip velocity
$\boldsymbol{u}_{S}$
on an equivalent (smooth) surface in the form
$\boldsymbol{u}_{S}=\unicode[STIX]{x1D716}{\mathcal{L}}\boldsymbol{ : }{\mathcal{E}}$
, where the dimensionless parameter
$\unicode[STIX]{x1D716}$
is a measure of the roughness amplitude,
${\mathcal{E}}$
denotes the strain-rate tensor associated with the outer flow in the vicinity of the surface and
${\mathcal{L}}$
is a third-order slip tensor arising from the microscopic geometry characterizing the rough surface. This boundary condition represents the tensorial generalization of the classical Navier slip condition. We derive this condition, in the limit of small microscopic Reynolds numbers, using a multi-scale technique that yields a closed system of equations, the solution of which allows the slip tensor to be univocally calculated, once the roughness geometry is specified. We validate this generalized slip condition by considering the flow about a rough sphere, the surface of which is covered with a hexagonal lattice of cylindrical protrusions. Comparisons with direct numerical simulations performed in both laminar and turbulent regimes allow us to assess the validity and limitations of this condition and of the mathematical model underlying the determination of the slip tensor
${\mathcal{L}}$
.

This paper addresses the integral energy fluxes in natural and controlled turbulent channel flows, where active skin-friction drag reduction techniques allow a more efficient use of the available power. We study whether the increased efficiency shows any general trend in how energy is dissipated by the mean velocity field (mean dissipation) and by the fluctuating velocity field (turbulent dissipation). Direct numerical simulations (DNS) of different control strategies are performed at constant power input (CPI), so that at statistical equilibrium, each flow (either uncontrolled or controlled by different means) has the same power input, hence the same global energy flux and, by definition, the same total energy dissipation rate. The simulations reveal that changes in mean and turbulent energy dissipation rates can be of either sign in a successfully controlled flow. A quantitative description of these changes is made possible by a new decomposition of the total dissipation, stemming from an extended Reynolds decomposition, where the mean velocity is split into a laminar component and a deviation from it. Thanks to the analytical expressions of the laminar quantities, exact relationships are derived that link the achieved flow rate increase and all energy fluxes in the flow system with two wall-normal integrals of the Reynolds shear stress and the Reynolds number. The dependence of the energy fluxes on the Reynolds number is elucidated with a simple model in which the control-dependent changes of the Reynolds shear stress are accounted for via a modification of the mean velocity profile. The physical meaning of the energy fluxes stemming from the new decomposition unveils their inter-relations and connection to flow control, so that a clear target for flow control can be identified.

This experimental study investigates the flow field and properties of a sweeping jet emitted from a fluidic oscillator into a quiescent environment. The aspect ratio of the outlet throat is 1. Stereoscopic particle image velocimetry is employed to measure the velocity field plane-by-plane. Simultaneously acquired pressure measurements provide a reference for phase correlating the individual planes yielding three-dimensional, time-resolved velocity information. Lagrangian and Eulerian visualization techniques illustrate the phase-averaged flow field. Circular head vortices, similar to the starting vortex of a steady jet, are formed repetitively when the jet is at its maximum deflection. The quantitative jet properties are determined from instantaneous velocity data using a cylindrical coordinate system that takes into account the changing deflection angle of the jet. The jet properties vary throughout the oscillation cycle. The maximum jet velocity decays much faster than that of a comparable steady jet indicating a higher momentum transfer to the environment. The entrainment rate of the spatially oscillating jet is larger than for a steady jet by a factor of 4. Most of the mass flow is entrained from the direction normal to the oscillation plane, which is accompanied by a significant increase in jet depth compared to a steady jet. The high entrainment rate results from the enlarged contact area between jet and ambient fluid due to the spatial oscillation. The jet’s total force exceeds that of an idealized steady jet by up to 30 %. The results are independent of the investigated oscillation frequencies in the range from 5 to 20 Hz.

Active drag reduction of an Ahmed body with a slant angle of
$25^{\circ }$
, corresponding to the high-drag regime, has been experimentally investigated at Reynolds number
$Re=1.7\times 10^{5}$
, based on the square root of the model cross-sectional area. Four individual actuations, produced by steady blowing, are applied separately around the edges of the rear window and vertical base, producing a drag reduction of up to 6–14 %. However, the combination of the individual actuations results in a drag reduction 29 %, higher than any previous drag reductions achieved experimentally and very close to the target (30 %) set by automotive industries. Extensive flow measurements are performed, with and without control, using force balance, pressure scanner, hot-wire, flow visualization and particle image velocimetry techniques. A marked change in the flow structure is captured in the wake of the body under control, including the flow separation bubbles, over the rear window or behind the vertical base, and the pair of C-pillar vortices at the two side edges of the rear window. The change is linked to the pressure rise on the slanted surface and the base. The mechanisms behind the effective control are proposed. The control efficiency is also estimated.

Designing effective control for complex three-dimensional flow fields proves to be non-trivial. Often, intuitive control strategies lead to suboptimal control. To navigate the control space, we use a linear parabolized stability analysis to guide the design of a control scheme for a trailing vortex flow field aft of a NACA0012 half-wing at an angle of attack
$\unicode[STIX]{x1D6FC}=5^{\circ }$
and a chord-based Reynolds number
$Re=1000$
. The stability results show that the unstable mode with the smallest growth rate (fifth wake mode) provides a pathway to excite a vortex instability, whereas the principal unstable mode does not. Inspired by this finding, we perform direct numerical simulations that excite each mode with body forces matching the shape function from the stability analysis. Relative to the uncontrolled case, the controlled flows show increased attenuation of circulation and peak streamwise vorticity, with the fifth-mode-based control set-up outperforming the principal-mode-based set-up. From these results, we conclude that a rudimentary linear stability analysis can provide key insights into the underlying physics and help engineers design effective physics-based flow control strategies.

We report on a numerical study of the vortex structure modifications and drag reduction in a flow over a rotationally oscillating circular cylinder at a high subcritical Reynolds number,
$Re=1.4\times 10^{5}$
. Considered are eight forcing frequencies
$f=f_{e}/f_{0}=0.5$
,
$1$
,
$1.5$
,
$2$
,
$2.5$
,
$3$
,
$4$
,
$5$
and three forcing amplitudes
$\unicode[STIX]{x1D6FA}=\unicode[STIX]{x1D6FA}_{e}D/2U_{\infty }=1$
,
$2$
,
$3$
, non-dimensionalized with
$f_{0}$
, which is the natural vortex-shedding frequency without forcing,
$U_{\infty }$
the free-stream velocity,
$D$
the diameter of the cylinder. In order to perform a parametric study of a large number of cases (
$24$
in total) with affordable computational resources, the three-dimensional unsteady computations were performed using a wall-integrated (WIN) second-moment (Reynolds-stress) Reynolds-averaged Navier–Stokes (RANS) turbulence closure, verified and validated by a dynamic large-eddy simulations (LES) for selected cases (
$f=2.5$
,
$\unicode[STIX]{x1D6FA}=2$
and
$f=4$
,
$\unicode[STIX]{x1D6FA}=2$
), as well as by the earlier LES and experiments of the flow over a stagnant cylinder at the same
$Re$
number described in Palkin et al. (Flow Turbul. Combust., vol. 97 (4), 2016, pp. 1017–1046). The drag reduction was detected at frequencies equal to and larger than
$f=2.5$
, while no reduction was observed for the cylinder subjected to oscillations with the natural frequency, even with very different values of the rotation amplitude. The maximum reduction of the drag coefficient is 88 % for the highest tested frequency
$f=5$
and amplitude
$\unicode[STIX]{x1D6FA}=2$
. However, a significant reduction of 78 % appears with the increase of
$f$
already for
$f=2.5$
and
$\unicode[STIX]{x1D6FA}=2$
. Such a dramatic reduction in the drag coefficient is the consequence of restructuring of the vortex-shedding topology and a markedly different pressure field featured by a shrinking of the low pressure region behind the cylinder, all dictated by the rotary oscillation. Despite the need to expend energy to force cylinder oscillations, the considered drag reduction mechanism seems a feasible practical option for drag control in some applications for
$Re>10^{4}$
, since the calculated power expenditure for cylinder oscillation under realistic scenarios is several times smaller than the power saved by the drag reduction.

We consider feedback flow control of the linearised complex Ginzburg–Landau system. The particular focus is on any trade-offs present when placing a single sensor and a single actuator. The work is presented in three parts. First, we consider the estimation problem in which a single sensor is used to estimate the entire flow field (without any control). Second, we consider the full information control problem in which the entire flow field is known, but only a single actuator is available for control. By considering the optimal sensor placement and optimal actuator placement while varying the stability of the system, a fundamental trade-off for both problems is made clear. Third, we consider the overall feedback control problem in which only a single sensor is available for measurement; and only a single actuator is available for control. By varying the stability of the system, similar fundamental trade-offs are made clear. We discuss implications for effective feedback control with a single sensor and a single actuator and compare it to previous placement methods.

We consider the application of active flow control to modify the radial pressure distribution of a single-phase wall-normal vortex. The present flow is based on the Burgers vortex model but with a no-slip boundary condition prescribed along its symmetry plane. The wall-normal vortex serves as a model for vortices that emerge upstream of turbomachinery, such as pumps. This study characterizes the baseline vortex unsteadiness through numerical simulation and dynamic mode decomposition. The insights gained from the baseline flow are used to develop an active flow control technique with rotating zero-net-mass blowing and suction with the objective of modifying the core-pressure distribution. The effectiveness of the control strategy is demonstrated by achieving a widened vortex core with increased pressure. This change in the flow field weakens the local strength of the wall-normal vortex core, potentially inhibiting the formation of hollow-core vortices, commonly encountered in liquids.

Drag control using a newly developed spanwise opposed wall-jet forcing (SOJF) method is studied via direct numerical simulation of the incompressible Navier–Stokes equations in a turbulent channel flow (at the friction Reynolds numbers
$Re_{\unicode[STIX]{x1D70F}}=180$
and 550). SOJF is characterized by three control parameters: the forcing amplitude
$A^{+}$
, the spanwise spacing
$\unicode[STIX]{x1D706}^{+}$
and the wall-jet height
$y_{c}^{+}$
(
$+$
indicates viscous scaling). At
$Re_{\unicode[STIX]{x1D70F}}=180$
, notable drag reduction is achieved for wide ranges of
$A^{+}$
,
$\unicode[STIX]{x1D706}^{+}$
and
$y_{c}^{+}$
, with an optimal drag reduction of approximately 19 % found for
$A^{+}\approx 0.015$
,
$\unicode[STIX]{x1D706}^{+}\approx 1200$
and
$y_{c}^{+}\approx 30$
. The drag reduction results from mergers of numerous low-speed typical individual streaks together by the wall jets, so that the slope of the merged streak envelope and hence the streak strength are reduced below the critical values required for streak instability as well as for transient growth; consequently, the generation of drag inducing near-wall streamwise vortices is suppressed. Through analysis using the FIK identity (Fukagata et al.Phys. Fluids, vol. 14 (11), 2002, pp. L73–L76) in combination with the triple decomposition and the spanwise wavenumber spectrum of the Reynolds shear stress, we find that the control significantly decreases skin friction due to the small scale random turbulent structures (from 75 to 23 % for the optimal case), but injects a dominant contribution at the forcing scale (approximately 34 %). As
$A^{+}$
or
$y_{c}^{+}$
increases, the drag reduction degrades due to the downwash near the initiation of the forcing wall jet. The energy input required for the excitation is found to be small, yielding a 17 % net power saving for the optimal control case. To determine the
$Re$
dependence of the drag reduction, the control strategy is further validated at a higher
$Re_{\unicode[STIX]{x1D70F}}=550$
. If the control parameters are kept the same as at
$Re_{\unicode[STIX]{x1D70F}}=180$
(i.e.
$A^{+}\approx 0.015$
,
$\unicode[STIX]{x1D706}^{+}\approx 1200$
,
$y_{c}^{+}\approx 30$
), the drag reduction decreases to 10 %; however, interestingly, with modestly changed parameters (
$A^{+}\approx 0.018$
,
$\unicode[STIX]{x1D706}^{+}\approx 1700$
,
$y_{c}^{+}\approx 50$
), drag reduction increases to about 15 %. This additional drag reduction results from the further suppression of turbulent structures in the buffer and log regions. This result, therefore, suggests prospects for drag reduction at even higher
$Re$
via a proper choice of the SOJF parameters.

While it has been known that an afterbody (i.e. the structural part of a bluff body downstream of the flow separation points) plays an important role affecting the wake characteristics and even may change the nature of the flow-induced vibration (FIV) of a structure, the question of whether an afterbody is essential for the occurrence of one particular common form of FIV, namely vortex-induced vibration (VIV), still remains. This has motivated the present study to experimentally investigate the FIV of an elastically mounted forward- or backward-facing D-section (closed semicircular) cylinder over the reduced velocity range
$2.3\leqslant U^{\ast }\leqslant 20$
, where
$U^{\ast }=U/(f_{nw}D)$
. Here,
$U$
is the free-stream velocity,
$D$
the cylinder diameter and
$f_{nw}$
the natural frequency of the system in quiescent fluid (water). The normal orientation with the body’s flat surface facing upstream is known to be subject to another common form of FIV, galloping, while the reverse D-section with the body’s curved surface facing upstream, due to the lack of an afterbody, has previously been reported to be immune to VIV. The fluid–structure system was modelled on a low-friction air-bearing system in conjunction with a recirculating water channel facility to achieve a low mass ratio (defined as the ratio of the total oscillating mass to that of the displaced fluid mass). Interestingly, through a careful overall examination of the dynamic responses, including the vibration amplitude and frequency, fluid forces and phases, our new findings showed that the D-section exhibits a VIV-dominated response for
$U^{\ast }<10$
, galloping-dominated response for
$U^{\ast }>12.5$
, and a transition regime with a VIV–galloping interaction in between. Also observed for the first time were interesting wake modes associated with these response regimes. However, in contrast to previous studies at high Reynolds number (defined by
$Re=UD/\unicode[STIX]{x1D708}$
, with
$\unicode[STIX]{x1D708}$
the kinematic viscosity), which have showed that the D-section was subject to ‘hard’ galloping that required a substantial initial amplitude to trigger, it was observed in the present study that the D-section can gallop softly from rest. Surprisingly, on the other hand, it was found that the reverse D-section exhibits pure VIV features. Remarkable similarities were observed in a direct comparison with a circular cylinder of the same mass ratio, in terms of the onset
$U^{\ast }$
of significant vibration, the peak amplitude (only approximately 6 % less than that of the circular cylinder), and also the fluid forces and phases. Of most significance, this study shows that an afterbody is not essential for VIV at low mass and damping ratios.

Our previous research has shown that an active flow control strategy using two-dimensional (2-D) harmonic blowing and suction with properly chosen frequency and amplitude can significantly reduce the separation region, delay transition to turbulence and can even relaminarize the flow. How such effective flow control for transition delay and relaminarization is affected by free-stream turbulence (FST) remains an open question. In order to answer this question, highly resolved direct numerical simulations (DNS) are carried out where very low-amplitude isotropic FST fluctuations are introduced at the inflow boundary of the computational domain. With FST the effectiveness of the flow control is not diminished, and the extent of the separated flow region is reduced by the same amount as for the zero FST case. However, a striking difference observed in the DNS is the fact that in the presence of even very low levels of FST, the flow transitions shortly downstream of the reattachment location of the bubble, contrary to the case without FST. It appears that this different behaviour for even very small levels of FST is caused by an interaction between the high-amplitude 2-D disturbances introduced by the flow control forcing and 3-D Klebanoff modes (K-modes) that are generated by the FST. The streamwise elongated streaks due to the K-modes cause a spanwise-periodic modulation of the basic flow and subsequently of the primary 2-D wave. The disturbances associated with this modulation exhibit strong growth and initiate the breakdown process to turbulence. Linear secondary instability investigations with respect to low-frequency 3-D disturbances are carried out based on the linearized Navier–Stokes equations. The response of the forced flow to the low-frequency 3-D disturbances reveals that the time-periodic base flow is unstable with respect to a wide range of 3-D modes. In particular, the wavelength associated with the spanwise spacing of the K-mode falls into the range of, and is in fact very close to, the most unstable 3-D disturbances. Results from the secondary instability analysis and the comparison with DNS results, support the conjecture that the forcing amplitude has a major impact on the onset and amplification rate of the K-modes: lowering the forcing amplitude postpones the onset of the growth of the K-modes and reduces the growth rate of the K-modes for a given FST intensity. The net effect of these two events is a delay of the transition onset. Nevertheless, the instability mechanism that governs the transition process is the same as previously identified, i.e. interaction of the K-mode and 2-D primary wave. Furthermore, for low levels of FST, the amplitude of the low-frequency K-modes scales linearly with the FST intensity in the approach boundary layer up to the secondary instability regime.

We consider a passive zero-mean scalar field organised into two layers of different concentrations in a three-dimensional plane channel flow subjected to a constant along-stream pressure gradient. We employ a nonlinear direct-adjoint-looping method to identify the optimal initial perturbation of the velocity field with given initial energy which yields ‘maximal’ mixing by a target time horizon, where maximal mixing is defined here as the minimisation of the spatially integrated variance of the concentration field. We verify in three-dimensional flows the conjecture by Foures et al. (J. Fluid Mech., vol. 748, 2014, pp. 241–277) that the initial perturbation which maximises the time-averaged energy gain of the flow leads to relatively weak mixing, and is qualitatively different from the optimal initial ‘mixing’ perturbation which exploits classical Taylor dispersion. We carry out the analysis for two different Reynolds numbers (
$Re=U_{m}h/\unicode[STIX]{x1D708}=500$
and
$Re=3000$
, where
$U_{m}$
is the maximum flow speed of the unperturbed flow,
$h$
is the channel half-depth and
$\unicode[STIX]{x1D708}$
is the kinematic viscosity of the fluid) demonstrating that this key finding is robust with respect to the transition to turbulence. We also identify the initial perturbations that minimise, at chosen target times, the ‘mix-norm’ of the concentration field, i.e. a Sobolev norm of negative index in the class introduced by Mathew et al. (Physica D, vol. 211, 2005, pp. 23–46). We show that the ‘true’ variance-based mixing strategy can be successfully and practicably approximated by the mix-norm minimisation since we find that the mix-norm-optimal initial perturbations are far less sensitive to changes in the target time horizon than their optimal variance-minimising counterparts.

A wind tunnel study has been carried out to investigate flow control around a hollow circular cylinder using passive jets driven by naturally occurring pressure differences. Flow enters the cylinder through spanwise holes along the stagnation line and exits through a spanwise distribution of holes at
$\pm 65^{\circ }$
. The diameter of the entry and exit holes were 1 % and 0.5 % of the cylinder diameter, respectively. Reynolds numbers were at the upper end of the subcritical regime and ranged from
$3\times 10^{4}$
to
$2.8\times 10^{5}$
. Jet spacings of 10 % and 20 % of the cylinder diameter were investigated, and the ratio of the average jet exit velocity to the cross-flow velocity at the boundary layer edge was found to rise to approximately 0.35 and 0.4, respectively, above a Reynolds number of
$1.5\times 10^{5}$
. Findings based on using the surface oil flow technique revealed a repeating, organised cellular pattern downstream of adjacent jet exit holes consisting of a primary counter-rotating vortex pair structure, followed by a secondary weaker pair. Downstream of adjacent exit holes, and centred midway between them, there exists a separation bubble which delays final flow separation compared with the flow directly downstream of a jet. The variation in the angular position of boundary layer separation across the span had the effect of suppressing von Kármán vortex shedding. This resulted in a drag coefficient, at the upper end of the Reynolds-number range studied, 14.5 % lower than that found using trip wires to initiate boundary layer transition.

A parametric study is conducted for the control of a turbulent jet using a single unsteady minijet. A number of control parameters that influence the decay rate
$K$
of the jet centreline mean velocity are investigated, including the mass flow rate ratio
$C_{m}$
, excitation frequency ratio
$f_{e}/f_{0}$
and exit diameter ratio
$d/D$
of the minijet to main jet, along with the duty cycle (
$\unicode[STIX]{x1D6FC}$
) of the minijet injection. Extensive hot-wire, particle image velocimetry and flow visualization measurements were performed in the manipulated jet. Various flow structures have been identified, such as the flapping flow, non-flapping flow and that showing a manipulable thrust vector, depending on
$C_{m}$
,
$f_{e}/f_{0}$
and
$\unicode[STIX]{x1D6FC}$
. Empirical scaling analysis unveils that, prior to the minijet impingement upon the wall of the nozzle and the generation of turbulence, the relationship
$K=g_{1}$
(
$C_{m}$
,
$f_{e}/f_{0}$
,
$d/D$
,
$\unicode[STIX]{x1D6FC}$
) may be reduced to
$K=g_{2}$
(
$\unicode[STIX]{x1D709}$
), where
$g_{1}$
and
$g_{2}$
are different functions and the scaling factor
$\unicode[STIX]{x1D709}=(\sqrt{MR}/\unicode[STIX]{x1D6FC})(d/D)^{n}$
(
$\sqrt{MR}\equiv C_{m}(D/d)$
is the momentum ratio and
$n$
is a constant that depends on
$\unicode[STIX]{x1D6FC}$
) is physically the effective momentum ratio per pulse or effective penetration depth. Discussion is conducted based on
$K=g_{2}$
(
$\unicode[STIX]{x1D709}$
), which provides important insight into the jet control physics.

The problem of transonic flow past an array of micro-electro-mechanical-type (MEMS-type) heating elements placed on a flat surface is investigated using the triple-deck theory. The compressible Navier–Stokes equations supplemented by the energy equation are considered for large Reynolds numbers. The triple-deck problem is formulated with the aid of the method of matched expansions. The resulting nonlinear viscous lower deck problem, coupled with the upper deck problem governed by the nonlinear Kármán–Guderley equation, is solved using a numerical method based on Chebyshev collocation and finite differences. Our results show the differences in subsonic and supersonic flow behaviour over heated elements. The results indicate the possibility of using the elements to favourably control the transonic flow field.

The predictive control of the self-sustained single spiral vortex breakdown mode is addressed in the three-dimensional flow geometry of Ruith et al. (2003) for a constant swirl number
$S=1.095$
. Based on adjoint optimization algorithms, two different control strategies have been designed. First, a quadratic objective function minimizing the radial velocity intensity, taking advantage of the physical mechanism underpinning spiral vortex breakdown. The second strategy focuses on the hydrodynamic instability properties using as objective function the growth rate of the most unstable global eigenmode. These minimization algorithms seek for an optimal volume force in an axisymmetric domain avoiding therefore expensive three-dimensional computations. In addition to considering eigenvalues around the base flow, we also investigate the stability around the mean flow and we find that it correctly predicts the frequency of the self-sustained single spiral vortex breakdown mode for Reynolds numbers up to
$Re=500$
. Close to the instability threshold, at a Reynolds value of
$Re=180$
, all these control strategies successfully quench the spiral vortex breakdown. The related volume force is found identical for the base and mean flow eigenvalue control even if the uncontrolled growth rates differ significantly. The control of the least unstable eigenvalue of the mean flow is not only found optimal at
$Re=180$
, it also stabilizes the flow at a Reynolds value as large as
$Re=300$
, which opens promising extensions to industrial applications.